------------------------------------------------------------------------------ -- -- -- GNAT COMPILER COMPONENTS -- -- -- -- E V A L _ F A T -- -- -- -- B o d y -- -- -- -- Copyright (C) 1992-2014, Free Software Foundation, Inc. -- -- -- -- GNAT is free software; you can redistribute it and/or modify it under -- -- terms of the GNU General Public License as published by the Free Soft- -- -- ware Foundation; either version 3, or (at your option) any later ver- -- -- sion. GNAT is distributed in the hope that it will be useful, but WITH- -- -- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY -- -- or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License -- -- for more details. You should have received a copy of the GNU General -- -- Public License distributed with GNAT; see file COPYING3. If not, go to -- -- http://www.gnu.org/licenses for a complete copy of the license. -- -- -- -- GNAT was originally developed by the GNAT team at New York University. -- -- Extensive contributions were provided by Ada Core Technologies Inc. -- -- -- ------------------------------------------------------------------------------ with Einfo; use Einfo; with Errout; use Errout; with Sem_Util; use Sem_Util; package body Eval_Fat is Radix : constant Int := 2; -- This code is currently only correct for the radix 2 case. We use the -- symbolic value Radix where possible to help in the unlikely case of -- anyone ever having to adjust this code for another value, and for -- documentation purposes. -- Another assumption is that the range of the floating-point type is -- symmetric around zero. type Radix_Power_Table is array (Int range 1 .. 4) of Int; Radix_Powers : constant Radix_Power_Table := (Radix ** 1, Radix ** 2, Radix ** 3, Radix ** 4); ----------------------- -- Local Subprograms -- ----------------------- procedure Decompose (RT : R; X : T; Fraction : out T; Exponent : out UI; Mode : Rounding_Mode := Round); -- Decomposes a non-zero floating-point number into fraction and exponent -- parts. The fraction is in the interval 1.0 / Radix .. T'Pred (1.0) and -- uses Rbase = Radix. The result is rounded to a nearest machine number. -------------- -- Adjacent -- -------------- function Adjacent (RT : R; X, Towards : T) return T is begin if Towards = X then return X; elsif Towards > X then return Succ (RT, X); else return Pred (RT, X); end if; end Adjacent; ------------- -- Ceiling -- ------------- function Ceiling (RT : R; X : T) return T is XT : constant T := Truncation (RT, X); begin if UR_Is_Negative (X) then return XT; elsif X = XT then return X; else return XT + Ureal_1; end if; end Ceiling; ------------- -- Compose -- ------------- function Compose (RT : R; Fraction : T; Exponent : UI) return T is Arg_Frac : T; Arg_Exp : UI; pragma Warnings (Off, Arg_Exp); begin Decompose (RT, Fraction, Arg_Frac, Arg_Exp); return Scaling (RT, Arg_Frac, Exponent); end Compose; --------------- -- Copy_Sign -- --------------- function Copy_Sign (RT : R; Value, Sign : T) return T is pragma Warnings (Off, RT); Result : T; begin Result := abs Value; if UR_Is_Negative (Sign) then return -Result; else return Result; end if; end Copy_Sign; --------------- -- Decompose -- --------------- procedure Decompose (RT : R; X : T; Fraction : out T; Exponent : out UI; Mode : Rounding_Mode := Round) is Int_F : UI; begin Decompose_Int (RT, abs X, Int_F, Exponent, Mode); Fraction := UR_From_Components (Num => Int_F, Den => Machine_Mantissa_Value (RT), Rbase => Radix, Negative => False); if UR_Is_Negative (X) then Fraction := -Fraction; end if; return; end Decompose; ------------------- -- Decompose_Int -- ------------------- -- This procedure should be modified with care, as there are many non- -- obvious details that may cause problems that are hard to detect. For -- zero arguments, Fraction and Exponent are set to zero. Note that sign -- of zero cannot be preserved. procedure Decompose_Int (RT : R; X : T; Fraction : out UI; Exponent : out UI; Mode : Rounding_Mode) is Base : Int := Rbase (X); N : UI := abs Numerator (X); D : UI := Denominator (X); N_Times_Radix : UI; Even : Boolean; -- True iff Fraction is even Most_Significant_Digit : constant UI := Radix ** (Machine_Mantissa_Value (RT) - 1); Uintp_Mark : Uintp.Save_Mark; -- The code is divided into blocks that systematically release -- intermediate values (this routine generates lots of junk). begin if N = Uint_0 then Fraction := Uint_0; Exponent := Uint_0; return; end if; Calculate_D_And_Exponent_1 : begin Uintp_Mark := Mark; Exponent := Uint_0; -- In cases where Base > 1, the actual denominator is Base**D. For -- cases where Base is a power of Radix, use the value 1 for the -- Denominator and adjust the exponent. -- Note: Exponent has different sign from D, because D is a divisor for Power in 1 .. Radix_Powers'Last loop if Base = Radix_Powers (Power) then Exponent := -D * Power; Base := 0; D := Uint_1; exit; end if; end loop; Release_And_Save (Uintp_Mark, D, Exponent); end Calculate_D_And_Exponent_1; if Base > 0 then Calculate_Exponent : begin Uintp_Mark := Mark; -- For bases that are a multiple of the Radix, divide the base by -- Radix and adjust the Exponent. This will help because D will be -- much smaller and faster to process. -- This occurs for decimal bases on machines with binary floating- -- point for example. When calculating 1E40, with Radix = 2, N -- will be 93 bits instead of 133. -- N E -- ------ * Radix -- D -- Base -- N E -- = -------------------------- * Radix -- D D -- (Base/Radix) * Radix -- N E-D -- = --------------- * Radix -- D -- (Base/Radix) -- This code is commented out, because it causes numerous -- failures in the regression suite. To be studied ??? while False and then Base > 0 and then Base mod Radix = 0 loop Base := Base / Radix; Exponent := Exponent + D; end loop; Release_And_Save (Uintp_Mark, Exponent); end Calculate_Exponent; -- For remaining bases we must actually compute the exponentiation -- Because the exponentiation can be negative, and D must be integer, -- the numerator is corrected instead. Calculate_N_And_D : begin Uintp_Mark := Mark; if D < 0 then N := N * Base ** (-D); D := Uint_1; else D := Base ** D; end if; Release_And_Save (Uintp_Mark, N, D); end Calculate_N_And_D; Base := 0; end if; -- Now scale N and D so that N / D is a value in the interval [1.0 / -- Radix, 1.0) and adjust Exponent accordingly, so the value N / D * -- Radix ** Exponent remains unchanged. -- Step 1 - Adjust N so N / D >= 1 / Radix, or N = 0 -- N and D are positive, so N / D >= 1 / Radix implies N * Radix >= D. -- As this scaling is not possible for N is Uint_0, zero is handled -- explicitly at the start of this subprogram. Calculate_N_And_Exponent : begin Uintp_Mark := Mark; N_Times_Radix := N * Radix; while not (N_Times_Radix >= D) loop N := N_Times_Radix; Exponent := Exponent - 1; N_Times_Radix := N * Radix; end loop; Release_And_Save (Uintp_Mark, N, Exponent); end Calculate_N_And_Exponent; -- Step 2 - Adjust D so N / D < 1 -- Scale up D so N / D < 1, so N < D Calculate_D_And_Exponent_2 : begin Uintp_Mark := Mark; while not (N < D) loop -- As N / D >= 1, N / (D * Radix) will be at least 1 / Radix, so -- the result of Step 1 stays valid D := D * Radix; Exponent := Exponent + 1; end loop; Release_And_Save (Uintp_Mark, D, Exponent); end Calculate_D_And_Exponent_2; -- Here the value N / D is in the range [1.0 / Radix .. 1.0) -- Now find the fraction by doing a very simple-minded division until -- enough digits have been computed. -- This division works for all radices, but is only efficient for a -- binary radix. It is just like a manual division algorithm, but -- instead of moving the denominator one digit right, we move the -- numerator one digit left so the numerator and denominator remain -- integral. Fraction := Uint_0; Even := True; Calculate_Fraction_And_N : begin Uintp_Mark := Mark; loop while N >= D loop N := N - D; Fraction := Fraction + 1; Even := not Even; end loop; -- Stop when the result is in [1.0 / Radix, 1.0) exit when Fraction >= Most_Significant_Digit; N := N * Radix; Fraction := Fraction * Radix; Even := True; end loop; Release_And_Save (Uintp_Mark, Fraction, N); end Calculate_Fraction_And_N; Calculate_Fraction_And_Exponent : begin Uintp_Mark := Mark; -- Determine correct rounding based on the remainder which is in -- N and the divisor D. The rounding is performed on the absolute -- value of X, so Ceiling and Floor need to check for the sign of -- X explicitly. case Mode is when Round_Even => -- This rounding mode corresponds to the unbiased rounding -- method that is used at run time. When the real value is -- exactly between two machine numbers, choose the machine -- number with its least significant bit equal to zero. -- The recommendation advice in RM 4.9(38) is that static -- expressions are rounded to machine numbers in the same -- way as the target machine does. if (Even and then N * 2 > D) or else (not Even and then N * 2 >= D) then Fraction := Fraction + 1; end if; when Round => -- Do not round to even as is done with IEEE arithmetic, but -- instead round away from zero when the result is exactly -- between two machine numbers. This biased rounding method -- should not be used to convert static expressions to -- machine numbers, see AI95-268. if N * 2 >= D then Fraction := Fraction + 1; end if; when Ceiling => if N > Uint_0 and then not UR_Is_Negative (X) then Fraction := Fraction + 1; end if; when Floor => if N > Uint_0 and then UR_Is_Negative (X) then Fraction := Fraction + 1; end if; end case; -- The result must be normalized to [1.0/Radix, 1.0), so adjust if -- the result is 1.0 because of rounding. if Fraction = Most_Significant_Digit * Radix then Fraction := Most_Significant_Digit; Exponent := Exponent + 1; end if; -- Put back sign after applying the rounding if UR_Is_Negative (X) then Fraction := -Fraction; end if; Release_And_Save (Uintp_Mark, Fraction, Exponent); end Calculate_Fraction_And_Exponent; end Decompose_Int; -------------- -- Exponent -- -------------- function Exponent (RT : R; X : T) return UI is X_Frac : UI; X_Exp : UI; pragma Warnings (Off, X_Frac); begin Decompose_Int (RT, X, X_Frac, X_Exp, Round_Even); return X_Exp; end Exponent; ----------- -- Floor -- ----------- function Floor (RT : R; X : T) return T is XT : constant T := Truncation (RT, X); begin if UR_Is_Positive (X) then return XT; elsif XT = X then return X; else return XT - Ureal_1; end if; end Floor; -------------- -- Fraction -- -------------- function Fraction (RT : R; X : T) return T is X_Frac : T; X_Exp : UI; pragma Warnings (Off, X_Exp); begin Decompose (RT, X, X_Frac, X_Exp); return X_Frac; end Fraction; ------------------ -- Leading_Part -- ------------------ function Leading_Part (RT : R; X : T; Radix_Digits : UI) return T is RD : constant UI := UI_Min (Radix_Digits, Machine_Mantissa_Value (RT)); L : UI; Y : T; begin L := Exponent (RT, X) - RD; Y := UR_From_Uint (UR_Trunc (Scaling (RT, X, -L))); return Scaling (RT, Y, L); end Leading_Part; ------------- -- Machine -- ------------- function Machine (RT : R; X : T; Mode : Rounding_Mode; Enode : Node_Id) return T is X_Frac : T; X_Exp : UI; Emin : constant UI := Machine_Emin_Value (RT); begin Decompose (RT, X, X_Frac, X_Exp, Mode); -- Case of denormalized number or (gradual) underflow -- A denormalized number is one with the minimum exponent Emin, but that -- breaks the assumption that the first digit of the mantissa is a one. -- This allows the first non-zero digit to be in any of the remaining -- Mant - 1 spots. The gap between subsequent denormalized numbers is -- the same as for the smallest normalized numbers. However, the number -- of significant digits left decreases as a result of the mantissa now -- having leading seros. if X_Exp < Emin then declare Emin_Den : constant UI := Machine_Emin_Value (RT) - Machine_Mantissa_Value (RT) + Uint_1; begin if X_Exp < Emin_Den or not Has_Denormals (RT) then if Has_Signed_Zeros (RT) and then UR_Is_Negative (X) then Error_Msg_N ("floating-point value underflows to -0.0??", Enode); return Ureal_M_0; else Error_Msg_N ("floating-point value underflows to 0.0??", Enode); return Ureal_0; end if; elsif Has_Denormals (RT) then -- Emin - Mant <= X_Exp < Emin, so result is denormal. Handle -- gradual underflow by first computing the number of -- significant bits still available for the mantissa and -- then truncating the fraction to this number of bits. -- If this value is different from the original fraction, -- precision is lost due to gradual underflow. -- We probably should round here and prevent double rounding as -- a result of first rounding to a model number and then to a -- machine number. However, this is an extremely rare case that -- is not worth the extra complexity. In any case, a warning is -- issued in cases where gradual underflow occurs. declare Denorm_Sig_Bits : constant UI := X_Exp - Emin_Den + 1; X_Frac_Denorm : constant T := UR_From_Components (UR_Trunc (Scaling (RT, abs X_Frac, Denorm_Sig_Bits)), Denorm_Sig_Bits, Radix, UR_Is_Negative (X)); begin if X_Frac_Denorm /= X_Frac then Error_Msg_N ("gradual underflow causes loss of precision??", Enode); X_Frac := X_Frac_Denorm; end if; end; end if; end; end if; return Scaling (RT, X_Frac, X_Exp); end Machine; ----------- -- Model -- ----------- function Model (RT : R; X : T) return T is X_Frac : T; X_Exp : UI; begin Decompose (RT, X, X_Frac, X_Exp); return Compose (RT, X_Frac, X_Exp); end Model; ---------- -- Pred -- ---------- function Pred (RT : R; X : T) return T is begin return -Succ (RT, -X); end Pred; --------------- -- Remainder -- --------------- function Remainder (RT : R; X, Y : T) return T is A : T; B : T; Arg : T; P : T; Arg_Frac : T; P_Frac : T; Sign_X : T; IEEE_Rem : T; Arg_Exp : UI; P_Exp : UI; K : UI; P_Even : Boolean; pragma Warnings (Off, Arg_Frac); begin if UR_Is_Positive (X) then Sign_X := Ureal_1; else Sign_X := -Ureal_1; end if; Arg := abs X; P := abs Y; if Arg < P then P_Even := True; IEEE_Rem := Arg; P_Exp := Exponent (RT, P); else -- ??? what about zero cases? Decompose (RT, Arg, Arg_Frac, Arg_Exp); Decompose (RT, P, P_Frac, P_Exp); P := Compose (RT, P_Frac, Arg_Exp); K := Arg_Exp - P_Exp; P_Even := True; IEEE_Rem := Arg; for Cnt in reverse 0 .. UI_To_Int (K) loop if IEEE_Rem >= P then P_Even := False; IEEE_Rem := IEEE_Rem - P; else P_Even := True; end if; P := P * Ureal_Half; end loop; end if; -- That completes the calculation of modulus remainder. The final step -- is get the IEEE remainder. Here we compare Rem with (abs Y) / 2. if P_Exp >= 0 then A := IEEE_Rem; B := abs Y * Ureal_Half; else A := IEEE_Rem * Ureal_2; B := abs Y; end if; if A > B or else (A = B and then not P_Even) then IEEE_Rem := IEEE_Rem - abs Y; end if; return Sign_X * IEEE_Rem; end Remainder; -------------- -- Rounding -- -------------- function Rounding (RT : R; X : T) return T is Result : T; Tail : T; begin Result := Truncation (RT, abs X); Tail := abs X - Result; if Tail >= Ureal_Half then Result := Result + Ureal_1; end if; if UR_Is_Negative (X) then return -Result; else return Result; end if; end Rounding; ------------- -- Scaling -- ------------- function Scaling (RT : R; X : T; Adjustment : UI) return T is pragma Warnings (Off, RT); begin if Rbase (X) = Radix then return UR_From_Components (Num => Numerator (X), Den => Denominator (X) - Adjustment, Rbase => Radix, Negative => UR_Is_Negative (X)); elsif Adjustment >= 0 then return X * Radix ** Adjustment; else return X / Radix ** (-Adjustment); end if; end Scaling; ---------- -- Succ -- ---------- function Succ (RT : R; X : T) return T is Emin : constant UI := Machine_Emin_Value (RT); Mantissa : constant UI := Machine_Mantissa_Value (RT); Exp : UI := UI_Max (Emin, Exponent (RT, X)); Frac : T; New_Frac : T; begin if UR_Is_Zero (X) then Exp := Emin; end if; -- Set exponent such that the radix point will be directly following the -- mantissa after scaling. if Has_Denormals (RT) or Exp /= Emin then Exp := Exp - Mantissa; else Exp := Exp - 1; end if; Frac := Scaling (RT, X, -Exp); New_Frac := Ceiling (RT, Frac); if New_Frac = Frac then if New_Frac = Scaling (RT, -Ureal_1, Mantissa - 1) then New_Frac := New_Frac + Scaling (RT, Ureal_1, Uint_Minus_1); else New_Frac := New_Frac + Ureal_1; end if; end if; return Scaling (RT, New_Frac, Exp); end Succ; ---------------- -- Truncation -- ---------------- function Truncation (RT : R; X : T) return T is pragma Warnings (Off, RT); begin return UR_From_Uint (UR_Trunc (X)); end Truncation; ----------------------- -- Unbiased_Rounding -- ----------------------- function Unbiased_Rounding (RT : R; X : T) return T is Abs_X : constant T := abs X; Result : T; Tail : T; begin Result := Truncation (RT, Abs_X); Tail := Abs_X - Result; if Tail > Ureal_Half then Result := Result + Ureal_1; elsif Tail = Ureal_Half then Result := Ureal_2 * Truncation (RT, (Result / Ureal_2) + Ureal_Half); end if; if UR_Is_Negative (X) then return -Result; elsif UR_Is_Positive (X) then return Result; -- For zero case, make sure sign of zero is preserved else return X; end if; end Unbiased_Rounding; end Eval_Fat;